The size is the number of ordered pairs of linearly independent vectors in a two-dimensional vector space over . For the first vector, there are possible choices (all nonzero vectors work). For the second vector, there are choices (all vectors that are not in the span of the first vector work). The product rule of combinatorics gives a total of possibilities.

This group is the quotient of by the subgroup of scalar matrices in . For a matrix to be in , we must have so . If (and hence ) is odd, then there are two solutions so the kernel has order two, giving . If is even (so ), then the kernel is trivial and .